Abstract-The papel' presents an analysis of the enel"!!;Y flux correspondin to propa atin modes in a circular wave uide consistin of uniaxial anisotropic ...
Proceedings
CEEM' 2012/Shang 'hai
Electromagnetic Waves in Perfect Electromagnetic Conductor Loaded Uniaxial Anisotropic Chiral Metamaterial Waveguide * M,A, Baqir and P,K. Choudhury Senior Member, IEEE Institute of Microengineering and Nanoelectronics (IMEN) Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor, Malaysia
'pankaj@ukm. my
Abstract-The
papel' presents an analysis of the enel"!!;Y flux
The present communication is devoted to the study of the
correspondin� to propa�atin� modes in a circular wave�uide
propagation of energy flux through a circular waveguide of
consistin� of uniaxial anisotropic chit-al metamaterial, the outer sUl'face of the �uide bein� assumed to be coated with a PEMC
(pel'fect electroma�netic conductor) medium. It is to be I'ecalled
that
the
case
of
PEMC
is
a
�enel'alization
of
the
cases
correspondin� to PEC (perfect electric conductor) and PMC
(pel'fect ma�netic conductOl'). The dispel'sion I'elation of the guide is derived by using suitable boundary conditions. The propa�ation
of
ne�ative
ener�y
flux
thl'ou�h
the
�uide
is
explored, which exists due to the backward wave propagation in
PEMC bounded uniaxial anisotropic chiral metamaterial.
uniaxial anisotropic chiral metamaterial. The outer surface of the guide is assumed to be coated with a PEMC (perfect electromagnetic conductor) medium, the kind of boundary introduced by Lindell and Sihvola [13]. The energy flux corresponding to two different kinds of uniaxial anisotropic mediums is investigated. The existence of negative energy flux is observed in the guide under consideration, which is attributed
to
the
been
extensively
simultaneously permeability
studied
negative
(/1),
values
chiral metamaterial, having radius
[1-6]. of
These
possess
(e)
permittivity
and
and exhibit negative reflection and refraction
which reverse Snell's law [1]. Due to such properties, these
polarized)
and
circularly polarized) waves. In Refs. explored
that,
when
the
chirality
the
RCP
(right
[2-4], it has been of
isotropic
chiral
metamaterial is increased enough, the circumstances related to negative
refraction
are
observed
the
corresponding
to
one
the outer surface of
guide;
time
dependency
factor
elwt
is
-z
axis of
suppressed
throughout the text. The constitutive relations corresponding
lGtl, GzUzuJ E-jK�Goflouzuz.H B = �fl, flzUzUJ H jK�GofloUzuz.E D
The phenomenon of negative refraction could also be
(left circularly
to
to uniaxial chiral metamaterials are prescribed as [14]
=
Here
K
(1a)
+
+
achieved by the use of chiral metamaterials. A plane wave upon incident on chiral metamaterials gets decomposed into
a,
1). A time harmonic incident wave propagates along the
nanotechnology such as superlensing and clocking [5,6].
LCP
due
which is coated with infinitely extended PEMC medium (fig.
materials find many superb applications in the field of
the
achieved
We consider a circular waveguide of uniaxial anisotropic
INTRODUCTION
Negatively indexed metamaterials are of great interest, and has
waves
II. ANAL YTICAL TREATMENT
Keyword� EM wave propagation, metamaterials, chiral fibers, I.
backward
anisotropic chiral metamaterial.
(lb)
+
defines the chirality parameter and liz is the unit vector
along the axis of the guide. Also, Gz, /1z and G/, /11 are the permittivity
and permeability of
the
medium
along
the
longitudinal and transversal directions, respectively, and the unit dyadic is written as
eigenwave - either for the LCP wave or for the RCP one. Within
the
metamaterials
context, have
been
circular of
waveguides
much
interest
of
chiral
among
the
researchers because of their many fascinating applications. Refs.
[7-10]
describe
the
propagation
behaviour
of
If the excited field in cylindrical coordinates is represented by Bessel function, i.e.
Ez = Jm(rr)eJm¢
electromagnetic waves through circular waveguides of chiral and chiral nihility metamaterials. In this stream, uniaxial
with
T=�k(;-j32;
anisotropic chiral metamaterials are of special kind, which are
ko
easy to realize artificially. In these materials, chirality appears
medium decomposes the wave into the (LCP,-) and the
only in one direction [11]. At microwave frequencies, uniaxial
(RCP,+) forms as
chiral metamaterials can be fabricated by using miniature spirals of conducting wires in host dielectric medium [11,12].
978-1-4673-0029-2/12/$26.00 ©2011 IEEE
being the free-space propagation constant, the chiral
E +z
104
-
AIn
J (T+r)e Jm ¢ HI
(2a)
Proceedings
CEEM' 2012/Shang 'hai
(2b) Here
Am and Bm are unknown coefficients with
that can take only discrete values, i.e.
m =
Now, the total field along the longitudinal component can be expressed as
m
as an integer
total field in chiral medium can be written as
=
+
-
E
(8a)
Hzl L[Ama+JJ, +r)+ BmajJTJ)]eJm¢e-J,Oz 17 t
(8b)
= =
E=E+ +E_
H 1.. {E 17
[AmJm(T+r)+BmJm(TJ)]ejnl¢e-j,BZ
Ezl
0,\,2,.... Now, the
(3a)
J
(3b)
Also, the transverse components of electromagnetic fields can be derived as
where 17 in the impedance of the chiral medium. P EM C m e d i um
Fig, I, TIIustration of the circular waveguide made of anisotropic chiral metamaterial with a PEMC boundary,
The electromagnetic field can be expressed in terms of transverse and longitudinal components as
E = (E{ +zEJe-J,Sz H = (Ht +zHz )e-J,Bz
(4a) (4b)
where fJ represents the axial phase constant. Now,
the
relation
between
the
transversal
and
the
longitudinal fields can be finally derived as (Sa)
(Sb)
In these equations, prime represents the differentiation with respect to the argument. Since the outer surface of the guide is coated
with
V' ,
=
V'
-
Z� oz
and
[
T� = A,z ez +flz et flt ,
±
[ eezt, flflzt, )2 +4K2 eeOtflfltO, ] (6) _
[ � Ez ) (�: - :: JJetflt /(kJeoflo)
(Ez,HJ = Ez,j
a=
and
17t
=
medium,
by
applying
+
using eq.
(9)
a,
the
suitable
we get the
(7)
105
=
and applying some tedious mathematical steps,
the energy flux can finally be derived as
fli V�·
978-1-4673-0029-2/12/$26.00 ©2011 IEEE
r
=
j3a_,�Jm (,_a){Jm-l (,+a)-Jm (,+a )}(I + M17,a+) -fJaJ!Jm (,+a)x {Jm-I (,_a) -Jm (,_a )}(I M'l,a_) 0 (10) In above eq. (10), M represents the admittance parameter. By
and the corresponding eigenfunctions will be [7]
where
PEMC
following dispersion relation:
It can be shown that the eigenvalues of eq. (2) will be of the form [7]
with
boundary conditions at the interface
CEEM' 2012/Shang 'hai
Proceedings
A Bm mfJa+fm (k+r)( mTpjm_1 (T+r) 2 ;4 77,r { -0.25fJk�Jm_1 (T])-Jm+l (T])) - 0.25vfJ2T:a_ xJm (TJ)(Jm_1 (T+r)-Jm+1 (T+r))}
exhibits negative flux whereas that for the Hll mode remains
+
positive. The negative flow of flux can be interpreted as the
/l.
Equation
backward wave propagation in the guide. Also, the flux becomes almost vanishing for these two modes after a radial
(11)
(11) represents the energy flux inside the circular
waveguide made of uniaxial anisotropic chiral metamaterial.
distance of about 5 /-lm. Corresponding to H-ll mode, a little amount of negative energy flux remains bounded near the interface of uniaxial chiral metamaterial and PEMC mediums. In the case of Type II medium, we find that, for all the three low-order modes of interest, the flux remains more confined in the central region of the guide. However, the
Ill. RESULTS AND DISCUSSION
trends of the flux patterns corresponding to Hll and H-ll
In the present analytical investigation, we focus the study
modes are very much changed along with the interesting
on the propagation of energy flux pattern through the guide
property that the flux for the H-ll mode now becomes positive,
corresponding to the allowed values of the propagation
i.e. the flux is now transported in the forward direction by this
constant
/3,
as
obtained
from
the
dispersion
relation
(10). We consider the operating wavelength and the guide has the radius 20 /-lm. As the
mode. As such, the selection of the type of medium remains of
represented by eq.
great importance in controlling the flux characteristics of the
to be 1.5 /-lm,
guide. We further observe that, with the Type II material
waveguide medium is assumed to be chiral in nature, E- and
composition, the variation of the energy flux corresponding to
H-fields remain coupled, resulting thereby the existence of
H11 and H-ll modes becomes linear with the radial distance
hybrid modes in the guide with right- and left-circular
with the Hll mode exhibiting a little higher amount of flux.
polaizations. We performed the investigation of the energy
The remarkable feature observed is that the energy flux
flux behaviour of these hybrid modes. For this purpose, we
supported by each of the modes becomes very small with
take into account two different types of uniaxial anisotropic
increasing radial dimension of the guide. As such, a very high
chiral metamaterials - viz. Type I and Type II. In the Type I medium, we assumed the permeability and permittivity values
amount of radial confinement of optical energy can be
=
flz
as
=
fll
flo, Gz
=
3xGo
and
GI
=
corresponding to the Type II medium are
3xGo and G, = -1.2xGo. Figures 2 and 3, respectively,
2xGo. Parameters flz = fll = flo, Gz =
achieved in such guides with uniaxial anisotropic chiral metamaterials bounded by PEMC medium. Sz (AU.) 4
illustrate the energy flux
patterns corresponding to the low-order modes in the guides with Type I and Type II kinds of material combinations. In these figures, dotted lines represent the flux behaviour of the lower order hybrid HOI modes, dashed lines correspond to the
5. x
situations of the HII modes, and the solid lines stand to
-2
demonstrate the case of H_11 mode. Sz
(AU.)
-4
4
.
. . ' .
..
6 1?:.
•••
••••
•
Ol.ll)O(ll········ �.M��I�········� .��002
r
(jiln)
.
Fig. 3. TIIustration of the energy flux in the guide with Type IT material composition.
I
2
�
o
r
I \
IV. CONCLUSIONS
' ... _ ---
The energy flux patterns through a circular waveguide of
..�. 5·. · -� ��.;:;� 6 .. . .. X IO
���� :=��;;:=�;.;e;;,;r��=(�02 r (pm)
.
uniaxial
-2
anisotropic
numerically
chiral
investigated.
It
metamaterial has
been
have
found
that
been the
propagation behaviour of the energy flux varies by changing the type of the uniaxial chiral medium inside the PEMC
-4
waveguide. The energy
material
combination,
the
energy
flux
remains
Interestingly, these two hybrid odes show
opposite trends of the flux characteristics, viz. the HOI mode
978-1-4673-0029-2/12/$26.00 ©2011 IEEE
as highly
backward waves by PEMC bounded guide with uniaxial
ACKNOWLEDGMENT
The authors are thankful to Prof. Burhanuddin Yeop Majlis,
prominent corresponding to the cases of HOI and Hll hybrid
2).
found
anisotropic chiral medium.
confined near the central region of the guide, which is more modes (fig.
been
negative energy flux remains as the evidence of supporting
We observe that, in the waveguide utilizing the Type I of
flux has
confined near the central region of guide. The propagation of
Fig. 2. Illustration of the energy flux in the guide with Type I material composition.
kind
. .. ..
.
the Director of IMEN (Universiti Kebangsaan Malaysia), for constant encouragement and help.
106
CEEM' 2012/Shang 'hai
Proceedings
[8]
REFERENCES [1] [2] [3]
[4]
[5] [6]
[7]
V.G. Veslago, "Electrodynamic substances with simultaneously negative values of E and �," Sov. Phy. Usp., vol. 10,pp. 509-514,1968. J.B. Pendry, "A chiral route to negative refraction," Science, vol. 306, pp. 1353-1355,2004. . S. Tretyakov, A. Sahvola, and L. Jylha, "Backward-wave regime and negative refraction in chiral composites," Photo and Nanost., vol. 3,pp. 107-115,2005. S. Zhang,Y.Park, J. Li, X. Lu, WZhang, and X. Zhang, "Negative refrective index in chiral metamaterials," Phy. Rev. Lett., vol. 102, pp. 023901.1 023901.4,2009. J.B. Pendry,"Negative refraction makes a perfect lens," Phy. Rev. Lett., vol. 85,pp. 3966-3969,2000. D. Schuring, JJ. Mock, BJ. Justice, S.A Cummer, J. B. Pendry, AF. Starr, and D.R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science, vol. 314,pp. 977-980,2006. J.F. Dong and J. Li,"Characteristics of guided modes in uniaxial chiral circular waveguide," Prog. in Electromagn. Res., vol. 124, pp. 331345,2012.
978-1-4673-0029-2/12/$26.00 ©2011 IEEE
[9] [10]
[11] [12]
[13] [14]
107
M.A. Baqir, A.A Syed and Q.A. Naqvi, "Electromagnetic waves in a circular waveguide of chiral nihility metamaterial," Prog. in Electromagn. Res. M, vol. 16,pp. 85-93,2011. P.K Choudhury and T. Yoshino, "Characterization of optical power confinement in a simple chirofiber," Optik, vol. 13,pp. 89-95,2002. KY. Lim, P.K Choudhury, and Z. Yusoff, "Chirofibers with helical windings - An analytical investigation," Optik, vol. 121, pp. 980-987, 2011 Q. Cheng and T..T. Cui, "Negative refraction in uniaxially anisotropic chiral media," Phy. Rev. B, vol. 73,pp. 113104-1-113104-4,2006. Q Cheng and TJ. Cui, "Refection and refraction properties of plane waves on the interface of uniaxially anisotropic chiral media," J. Opt. Soc. Am. A, vol. 23,pp. 3203-3207,2006. TV Lindell and AH. Sihvola,"Perfect electromagnetic conductor," J. Electromagn. Waves andAppl., vol. 19,2005,pp. 861-869. l.V. Lindell and AJ. Viitanen, "Plane wave propagation in uniaxial bianisotropic medium," Electron. Lett., vol. 29,pp. 150-152,1993.
